Is this the real definition of an ordinal number? I have to prove that if $\alpha$ is ordinal and $\beta \in \alpha$ then $\beta$ is oridinal.
To do this I wanted to prove that for every $x\in\beta$, $x\in\alpha$.
This seems like it should be true, but given the definition of ordinal numbers I have in my textbook, it doesn't seem so. It states the following
"A set $x$ is an ordinal number if $(a, \in)$ is transitive and well-ordered". Firstly, I'm pretty sure that a well-order implies transitivity, but that's not the problem here.
I can construct a set like $\{\emptyset,\{\emptyset,\{\emptyset\}\}\}$ which is well-ordered with the $\in$ relation, but $\{\emptyset\}\in\{\emptyset,\{\emptyset\}\}$ and $\{\emptyset,\{\emptyset\}\}\in\{\emptyset,\{\emptyset,\{\emptyset\}\}\}$ and $\{\emptyset\}\notin\{\emptyset,\{\emptyset,\{\emptyset\}\}\}$.
Is this definition wrong, or does this property really not hold for ordinals? Or maybe I'm just missing something.
 A: Your definition is correct and here is the proof you are looking for: Let $\alpha$ be an ordinal and let $\beta \in \alpha$. We show that $\beta$ has the two properties you mentioned:
Well-ordering: Let $X$ be a non-empty subset of $\beta$. From the transitivity of $\alpha$, it follows that $X \subseteq \alpha$ and since $\alpha$ is well-ordered, $X$ must contain a least element. Now, we show that the order $\in$ is total in $\beta$. Assume that $x,y \in \beta$. By the transitivity of $\alpha$, we have $x,y \in \alpha$ and therefore $x=y$, $x\in y$ or $y \in x$. Therefore, $\beta$ is a well-order.
Transitivity: Assume $x \in y \in \beta$. Then $x \in y \in \alpha$ and hence $x \in \alpha$. Since $\alpha$ is a well-order we must have $x=\beta$, $\beta \in x$ or $x \in \beta$ but the first two possibilities lead to a contradiction since $\beta$ is a well-order. Hence $x \in \beta$.
$\textbf{Edit:}$ Here is an inductive definition with which you can produce some ordinal numbers $\alpha_0,\alpha_1,...$:
$$\alpha_0=\emptyset$$ 
$$\alpha_{n+1}=\alpha_n \cup \{\alpha_n\}$$
You can verify that all these $\alpha_i$ are indeed ordinals. However, note that the above definition does not generate all ordinals.
Some examples: 
$$\alpha_0=\emptyset$$ 
$$\alpha_1=\{\emptyset\}$$ 
$$\alpha_2=\{\emptyset,\{\emptyset\}\}$$ 
$$\alpha_3=\{\emptyset,\{\emptyset\}, \{\emptyset,\{\emptyset\}\} \}$$
